# Mathematics minor requirements

My institution (in USA) is planning to introduce a minor in mathematics and here are the proposed course requirements:

• Calculus I
• Calculus II
• Calculus III or Differential Equations
• Any two of the following three: Linear Algebra, Bridge Course to Abstract Mathematics and Elementary Statistics II.

I personally feel uncomfortable with the proposal: In my humble understanding, Calculus and Linear Algebra are of fundamental importance to mathematics and I tend to believe that both Calculus III and Linear Algebra should be required for a mathematics minor. Differential Equations may be an elective; I do not consider Differential Equations as necessarily more important than Real Analysis, Complex Analysis, Abstract Algebra, Functional Analysis and Topology.

So my question is, what course are usually considered as required for minor in mathematics?

• No inclusion of Discrete Math, Graph Theory, Combinatorics, Abstract Algebra: Seems a narrow Calculus-oriented minor. – Joseph O'Rourke Apr 13 at 23:40
• It does not seem to be a question. Don't tell us, tell them. – Gerald Edgar Apr 13 at 23:40
• I think it wold be good to have something at the end of that bridge. – Andreas Blass Apr 14 at 1:00
• I always argued that the purpose of a math minor was in part to create interesting load hours for us math professors. This proposal is weak. What is the point of having a minor when it only maybe gets them to take 1 or 2 additional courses at that. At least one more course wouldn't hurt. It should be a course which is not a requirement of other common majors. – James S. Cook Apr 14 at 1:53
• One way a math minor can help a department is to bolster enrollments in courses taken by math majors, which might otherwise have low enrollment. But this is not your situation. I don’t know anything about your institution, but maybe you should be careful to not give too many electives in your minor. Otherwise, you might create a stressful situation with low enrolled classes. – user52817 Apr 14 at 2:25

### Comparison to other programs

While implementing a new minor, it might make sense to see what other institutions are doing. Thus it would behoove you to look at the catalogs of other institutions to see what they do. I would recommend looking at a few top-flight schools (UCLA, Stanford, Cornell, MIT, Michigan, etc) to get a sense of what they do, then look at several schools in the same tier as your own institution. If you don't have the faculty, students, or courses to offer something at least as good as institutions similar to your own, then you probably aren't doing your students any good by offering a minor in mathematics.

Making a comparison to institutions where I have knowlege, my impression of the proposed minor is that it seems a little anemic:

• At my bachelor and masters institution, University of Nevada, Reno, the requirements for a mathematics minor are three semesters of calculus (a year of single variable calculus, and a semester of multivariable calculus), followed by three semesters of upper division mathematics. The upper division courses may be chosen freely from among all offerings at the university, though a lower division "intro to proof writing" course is a prerequisite for many (though not all) of the upper division classes, so a typical mathematic minor ends up being something like:

• Calculus (12 semester credits)
• Intro to Proof Writing (3 semester credits)
• upper division coursework (9 semester credits)

This is a total of 24 semester credits, and requires (more or less) that a student take one mathematics course for each of six or seven semesters (i.e. if one takes only one mathematics course at each opportunity, the minor will take about three years to complete).

Alternatively, there is a statistics minor offered by the department, which has similar requirements, but is more narrowly focused.

• At my current institution, University of California Riverside (.pdf, see page 404), the requirements for a mathematics minor are five quarters of calculus (a year of single variable calculus and two quarters of multivariable calculus), followed by six quarters of upper division classes. Again, the upper division classes may be taken freely from among everything offered, though there are some restrictions (there is a limit on the number of "reading courses" which may be taken, and a limit on the number of courses which can be applied to another program in addition to the mathematics minor). Again, the "bridge course" is a lower division class, and does not count toward the minor, though is recommended before taking upper division classes. A typical course list is something like

• Single Variable Calculus (12 quarter credits)
• Multivariable Calculus (8 quarter credits)
• Introduction to Discrete Structures (the bridge course, 4 quarter credits)
• upper division coursework (24 quarter credits)

This is a total of 48 quarter credits, and requires (more or less) that a student take one mathematics course for each of twelve quarters (i.e. if one takes only one mathematics course at each opportunity, the minor will take about three years to complete).

By comparison, the proposed curriculum requires a year and a half of lower-division coursework (which is on par with UNR and UCR), but only a year of upper-division coursework (which is less than either UNR or UCR), and this extra year may include a bridge course (which is considered lower-division at the above cited institutions). The proposed minor is also quite narrow, and is likely to overlap significantly with coursework that is already being taken for other majors (for example, CS and physics majors are often required to take calculus, differential equations, and linear algebra).

### What is the goal?

In addition to comparing your proposed program to other programs, you might want to consider what your goal is. I am of the opinion that the goal of a minor in any field is to get students to take classes which they would not otherwise take. A cynical reason for this is to boost enrollment in classes which might not otherwise run, but which instructors are keen to teach. This is also a way of bolstering enrollment in courses for math majors, but this seems not to be a problem in the current context, as the asker's institution appears not to offer such a major. A less cynical (and perhaps more meaningful and important) rationale is to broaden the horizons of students by getting them to take courses which expose them to some of the depths of mathematics.

From this point of view, it seems reasonable to remove the bridge course from among those that count towards the minor. A bridge course is a prerequisite (perhaps) for upper division coursework, but is not, in and of itself, an upper-division class. It would also be reasonable to add something akin to "Only $$x$$ of the upper-division courses may count toward major requirements," which is a way of forcing students to expand their horizons a bit. If a student finishes their major and notices "Oh, hey! I completed a math minor, too! I should put that on my transcript!", then I think that you have failed to provide a compelling program. What you want students to say is "Oh, hey! If I take just one (or two) more math classes, then I can get a minor in math! I should do that!"

If you do not have the faculty, courses, or students to sustain these requirements, then it might not be worth offering a mathematics minor.

### TL;DR

• The lower-division offerings seem fine, and are inline with other institutions.

• Consider dropping the bridge course from the list of courses which count towards the minor, particularly if students are only going to be required to take two courses beyond their lower-division calculus / differential equations.

• Consider requiring three upper-division courses (rather than just two).

• Consider expanding the list of upper-division courses which will satisfy the minor.

• Consider adding a clause to the minor requirements limiting the number of courses which may be counted both towards a student's major and the mathematics minor.

• If your college or program are so small that you cannot sustain a mathematics minor which offers more upper-division courses which are not required for some other major, consider abandoning the idea of implementing a mathematics minor.

• "intro to proof writing" is a whole course in itself? Should not the students know how to write proofs from their HS geometry course? – Rusty Core Apr 17 at 0:02
• @RustyCore You have to take students where they are. Most high schools do not teach mathematical writing. In that sense, these kinds of classes are a form of remediation, which is why they are generally not counted toward a minor (or major), but recommended (or even required) for upper division classes. – Xander Henderson Apr 17 at 0:11

Consider an opposite perspective (unless you were only looking for affirmation). Maybe you learn something by getting challenged.

Differential equations has huge application in other fields. I'm fine with the ODE course at same level as the calc 3 course. Probably even prefer it if forced to pick. It's very similar to integral calculus (calc 2) in thought pattern and I've used it more than calc 3 as an engineer and scientist.

Linear Algebra is consistently overrated on this site by the theory brigade. I think it comes from I donno being more abstract or something. Yes, there's applications too (but then they're not interested in them either). And for a practical matter, most engineers/scientists can get the limited amount they need in a quick chapter on matrices, within a math methods course that mostly hits PDEs (time wise).

Some of the topics (Real Analysis, Functional Analysis, Complex Analysis) are relatively more advanced than ODEs and I think less important than basic ODEs. Also, I think the practice in series solution of ODEs is good to have prior to these courses.

In an ideal world, sure the kids would take a gazillion courses and be super advanced. But clearly this is a much more constrained situation than that. (Think linear programming haha.) So, I find the math minor proposed to be extremely practical and normal. I'm on their side and against your side.

Before you dismiss my scientist/engineer slant consider that they are also time constrained. So the courses that are key for them may also be key for time/teacher constrained students at remote Alpine Lutheran colleges. ;-)

• Your disparaging remarks about the "theory brigade" aside, if a student is majoring in engineering, then their program will likely require them to take many of the courses you suggest. The point of a minor, typically, is to expose students to courses which they would not have otherwise taken in order to broaden their horizons. As algebra (including linear), analysis (real, complex, etc), and number theory are pretty foundational to modern mathematics, it seems reasonable to require some of these for a minor. Depending on the strengths of the department, prob and stats is another way to go. – Xander Henderson Apr 14 at 3:44
• He's teaching at a seminary. Limited technical majors. If you read the comments, you'd know he doesn't even have a math major. I feel very justified in disparaging the theory brigade when I get a comment like yours that shows a quick rush to correct, based on lack of inspection of the overall situation. – guest Apr 14 at 13:30
• Oh...and if it turns out not to be exactly a seminary but just some Swiss version of a TX bible college, so what? The key point remains about the situation. (Anticipating the pedant comment that corrects a detail and ignores an insight.) – guest Apr 14 at 13:44
• A quarter loaf is better than none. Perfect is the enemy of better... – guest Apr 14 at 13:47